Page:EB1911 - Volume 28.djvu/443

The general solution of (1) was given by J. le R. d’Alembert in 1747: it is

$y=f(ct+x)+F(ct+x),\,$

(3)

where the functions f, F are arbitrary. The first term is unaltered in value when x and ct are increased by equal amounts; hence this term, taken by itself, represents a wave-form which is propagated without change in the direction of x-positive with the constant velocity c. The second term represents in like manner a wave-form travelling with the same velocity in the direction of x-negative; and the most general free motion of the string consists of two such wave-forms superposed. In the case of an initial disturbance confined to a finite portion of an unlimited string, the motion finally resolves itself into two waves travelling unchanged in opposite directions, in these separate waves we have

$y=\mp cy^{\prime },\,$

(4)

as appears from (3), or from simple geometrical considerations. It is to be noticed, in this as in all analogous cases, that the wave-velocity appears as the square root of the ratio of two quantities, one of which represents (in a generalized sense) the elasticity of the medians, and the other its inertia.

The expressions for the kinetic and potential energies of any portion of the string are

$T={\tfrac {1}{2}}\rho \int y^{2}dx,\quad V={\tfrac {1}{2}}P\int y^{\prime 2}dx$

(5)

where the integrations extend over the portion considered. The relation (4) shows that in a single progressive wave the total energy is half kinetic and half potential.

When a point of the string (say the origin O) is fixed, the solution takes the form

$y=f(ct-x)-f(ct+x).\,$

(6)

As applied (for instance) to the portion of the string to the left of O, this indicates the superposition of a reflected wave represented by the second term on the direct wave represented by the first. The reflected wave has the same amplitudes at corresponding points as the incident wave, as is indeed required by the principle of energy, but its sign is reversed.

The reflection of a wave at the junction of two strings of unequal densities ρ, ρ' is of interest on account of the optical analogy. If A, B be the ratios of the amplitudes us the reflected and transmitted waves, respectively, to the corresponding amplitudes in the incident wave, it is found that

$A=-(\mu -1)/(\mu +1),\quad B=2\mu /(\mu +1),\,$

(7)

where μ=√(ρ′/ρ), is the ratio of the wave-velocities. This is on the hypothesis of an abrupt change of density; if the transition be gradual there may be little or no reflection.

The theory of waves of longitudinal vibration in a uniform straight rod follows exactly the same lines. If ξ denote the displacement of a particle whose undisturbed position is x, the length of an element of the central line is altered from δx to δx+δξ, and the elongation is therefore measured by ξ'. The tension across any section is accordingly Eωξ', where ω is is the sectional area, and E denotes Young’s modulus for the material of the rod (see Elasticity). The rate of change of momentum of the portion included between two consecutive cross-sections is ρωδx.ξ, where ρ now stands for the volume-density. Equating this to the difference of the tensions on these sections we obtain

$\xi =c^{2}\xi ^{\prime \prime }$

(8)

where

$c={\sqrt {}}(E/\rho ).\,$

(9)

The solution and the interpretation are the same as in the case of (1). It may be noted that in an iron or steel rod the wave-velocity given by (9) amounts roughly to about five kilometres per second.

The theory of plane elastic waves in an unlimited medium, whether fluid or solid, leads to differential equations of exactly the same type. Thus in the case of a fluid medium, if the displacement normal to the wave-fronts be a function of t and x, only, the equation of motion of a thin stratum initially bounded by the planes x and x+δx is

$\rho _{0}{\frac {\partial ^{2}\xi }{\partial t^{2}}}=-{\frac {\partial p}{\partial x}},$

(10)

where p is the pressure, and ρ₀ the undisturbed density. If p depends only on the density, we may write, for small disturbances,

$p=p_{0}+ks,\,$

(11)

where s=(ρ−ρ₀)ρ₀, is the "condensation," and k is the coefficient of cubic elasticity. Since s=−d ξ/dx, this leads to

${\frac {\partial ^{2}\xi }{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}\xi }{\partial x^{2}}}\,$

(12)

with

$c={\sqrt {}}(k/p)\,$

(13)

The latter formula gives for the velocity of sound in water a value (about 1490 metres per second at 15° C.) which is in good agreement with direct observation. In the case of a gas, if we neglect variations of temperature, we have k=p₀ by Boyle’s Law, and therefore c=√p₀ / ρ₀. This result, which is due substantially to Sir I. Newton, gives, however, a value considerably below the true velocity of sound. The discrepancy was explained by P. S. Laplace (about 1806?). The temperature is not really constant, but rises and falls as the gas is alternately compressed and rarefied. When this is allowed for we have k=γp₀, where γ is the ratio of the two specific heats uf the gas, and therefore c=√(γp₀ / ρ₀). For air, γ=1·41, and the consequent value of c agrees well with the best direct determinations (332 metres per second at 0° C.).

The potential energy of a system of sound waves is 1/2ks² per unit volume. As in all cases of propagation in one dimension, the energy of a single progressive system is half kinetic and half potential.

In the case of an unlimited isotropic elastic solid medium two types of plane waves are possible, viz. the displacement may be normal or tangential to the wave-fronts. The axis of x being taken in the direction of propagation, then in the case of a normal displacement ξ the traction normal to the wave-front is (λ+2μ)∂ξ/∂x, where λ, μ are the elastic constants of the medium, viz. pi is the “rigidity,” and λ＝k2/3μ, where k is the cubic elasticity. This leads to the equation

ξ＝a2ξ″

(14)

a＝√(λ+2μ)/ρ}＝√{k+4/3μ)/ρ}

(15)

The wave-velocity is greater than in the case of the longitudinal vibrations of a rod, owing to the lateral yielding which takes place in the latter case. In the case of a displacement η parallel to the axis of y, and therefore tangential to the wave-fronts, we have a shearing strain ∂η/∂x, and a corresponding shearing stress μ∂η/∂x. This leads to

η..＝b2η

(16)

with

b＝√(μ/ρ).

(17)

In the case of steel (k＝1·841 . 10¹², μ = 8·19. 10¹¹, ρ = 7·849 C.G.S.) the wave- velocities a, b come out to be 6·1 and 3·2 kilometres per second, respectively.

If the medium be crystalline the velocity of propagation of plane waves will depend also on the aspect of the wave-front. For any given direction of the wave-normal there are in the most general case three distinct velocities of wave-propagation, each with its own direction of particle-vibration. These latter directions are perpendicular to each other, but in general oblique to the wave-front. For certain types of crystalline structure the results simplify, but it is unnecessary to enter into further details, as the matter is chiefly of interest in relation to the now abandoned elastic-solid theories of double-refraction. For the modern electric theory of light see Light, and Electric Waves.

Finally, it may be noticed that the conditions of wave-propagation without change of type may be investigated in another manner. If we impress on the whole medium a velocity equal and opposite to that of the wave we obtain a “steady” or “stationary” state in which the circumstances at any particular point of space are constant. Thus in the case of the vibrations of an inextensible string we may, in the first instance, imagine the string to run through a fixed smooth tube having the form of the wave. The velocity c being constant there is no tangential acceleration, and the tension P is accordingly uniform. The resultant of the tensions on the two ends of an element δs is Pδs/R, in the direction of the normal, where R denotes the radius of curvature. This will be exactly sufficient to produce the normal acceleration c²R in the mass ρδς, provided c²＝P/ρ. Under this condition the tube, which now exerts no pressure on the string, may be abolished, and we have a free stationary wave on a moving string. This argument is due to P. G. Tait.

The method was applied to the case of air-waves by W. J. M. Rankine in 1870. When a gas flows steadily through a straight tube of unit section, the mass m which crosses any section in unit time must be the same; hence if u be the velocity we have

ρu＝m

(18)

Again, the mass which at time t occupies the space between two fixed sections (which we will distinguish by suffixes) has its momentum increased in the time δt by (mu₂−mu₂) U, whence

p1−p2＝m(u2−u1)

(19)

Combined with (18) this gives

p1+m2/ρ1＝p2+m2/ρ2

(20)

Hence for absolutely steady motion it is essential that the expression p+m²/ρ should have the same value throughout the wave. This condition is not accurately fulfilled by any known substance, whether subject to the “isothermal” or “adiabatic” condition; but in the rase of small variations of pressure and density the relation is equivalent to

m2＝ρ2dp/dρ

(21)

and therefore by (18), if c denote the general velocity of the current,

c2＝dp/dρ＝k/ρ,

(22)

in agreement with (13). The fact that the condition (20) can only be satisfied approximately shows that some progressive change of type must inevitably take place in sound-waves of finite amplitude. This question has been examined by S. D. Poisson (1807), Sir G. G. Stokes (1848), B. Riemann (1858), S. Earnshaw (1858), W. J. M. Rankine (1870), Lord Rayleigh (1878) and others. It appears that